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The prime aim of this book is to focus on this matrix called ‘Jacket matrix’ which is generalized Hadamard matrix along with the other orthogonal/unitary matrices and their applications to orthogonal code design, wireless communications, signal processing: to create more efficient algorithms and for their better performance in the tasks assigned. For this purpose there are many other unitary matrices namely reciprocal DFT, DCT, Slant, Haar, Wavelet, MIMO precoding, block diagonalization zero-forcing and subspace diagonal channel matrix of interference alignment. The main property of the Jacket matrix is that its inverse can be obtained by its element (block)-wise inverse or block-wise diagonal inverse. The Jacket transform is derived by using Center weighted Hadamard transform corresponding to Hadamard matrix and some systematical matrices. We present the Jacket transform and a simple decomposition of its matrix, which is used to develop a fast algorithm for signal processing and communications. The matrix decomposition is of the form of the Kronecker product of identity matrices and successively lower order coefficient matrices